Iid guess it would have to be Tim Berner Lee's "domain" - but i would guess we are talking far before the WWW. There were domains then surely, but only serving email maybe and things like telnet and gopher.
As guess what about the National Science Federation? The US Defense department that "invented" the Internet?
If you do a search on google on "oldest domain name" there is a lot of claims but very little agreement.
Iid be following this with interest!
15 Jan 85 Begin New Domain RegistrationNew domains may register according to the procedures and
restrictions described in RFC-920 [5].
Which means that nordu.net was registered even before the registry was officially operative. I guess that's about as far back as you can go... ;)
All of the original edu domains I checked were registered a few months later, probably still running under the now extinct (except for reverse lookup) .arpa label until then.
Nordu.net is the academic backbone of the northern european countries (Scandinavia, Denmark, and Iceland). They have chosen a very speaking and catchy name for their purposes, which probably sounds even better in the scandinavian languages than it does in english.
In a way, this domain was a precursor to the current trends, where eg. Finland has one of the highest densities of mobile phones per inhabitants in the world. Those guys really have a solid tradition in high-tech communications.